Ons-sets and mutual absolute continuity of measures on homogeneous spaces
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We extend a recent result of A. Jonsson about mutual absolute continuity of twoD s -measures on ans-setF ⊂R n to the homogeneous spaces (X, d, μ) of Coifman, Weiss. Here we define Hausdorff measure, Hausdorff dimension,D s -set andd-set relative to the measureμ. Our main result holds for so called (s, d)-sets,d ≥s, and is stronger than Jonssons result even inR n . As applications we interpret this Hausdorff dimension as a relative dimension for very regular sets and show that it in general depends strongly onμ. For this purpose we construct a strictly increasing functionf :R →R, whose measure is doubling and concentrated on a set of arbitrary small Hausdorff dimension. The extension off to a quasiconformal map of the half plane onto itself sharpens a classical example of Ahlfors-Beurling.
KeywordsHomogeneous Space Hausdorff Dimension Besov Space Quasiconformal Mapping Hausdorff Measure
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